Turing Machines. Lecture 8

Transcription

1 Turing Machines Lecture 8 1

2 Course Trajectory We will see algorithms, what can be done. But what cannot be done? 2

3 Computation Problem: To compute a function F that maps each input (a string) to an output bit Program: A finitely described process taking a string as input, and outputting a bit (or not halting) P computes F if for every x, P(x) outputs F(x) and halts 3 A program is a finite bit string Programs can be enumerated listed sequentially (say, lexicographically) so that every program appears somewhere in the list The set of all programs is countable. 1 ε

4 Computation Problem: To compute a function F that maps each input (a string) to an output bit Program: A finitely described process taking a string as input, and outputting a bit (or not halting) P computes F if for every x, P(x) outputs F(x) and halts 4 A function assigns a bit to each finite string Corresponds to an infinite bit string The set of all functions is uncountable! As numerous as, say, real numbers in [0,1] 1 ε

5 Computation Problem: To compute a function F that maps each input (a string) to an output bit Program: A finitely described process taking a string as input, and outputting a bit (or not halting) P computes F if for every x, P(x) outputs F(x) and halts There are uncountably many functions! But only countably many programs 5 Almost every function is uncomputable! (non constructive proof)

6 Course Trajectory We will be looking at what can be computed at all? What cannot be decided (undecidability) 6

7 What about a particular problems? Given program P, input w: w P Will halt or run into an infinite loop? Halting problem! 7

8 Halting Problem Undecidable Given program P: P P Write a program that decides if a program halts. Dual view of program as program and as data. 8

9 Alas! There is no program that solves the Halting Problem! Way to view code as data allows diagonalization proof. 9

10 Computing What does it mean to compute something? There is no algorithm for the halting problem What does it mean for something to be an algorithm formally? Then I can say For all algorithms 10

11 A Brief History of Computing Leibnitz (circa 1600): Believed in universal language for encoding any problem (math, philosophy, religion). Thought if you properly encode any problem in binary form, there is a way to calculate an answer (run the algorithm) calculemus! 11

12 A Brief History of Computing Babbage (circa 1860): built Difference and Analytical Engine (Add, multiply, etc). It marks the transition from mechanised arithmetic to fully-fledged general purpose computation. Hypothesized that any mathematical question can be answered by the analytical engine, suitably encoded. 12

13 A Brief History of Computing Hilbert (circa 1900): What can we prove in mathematical world? Proving and computing almost identical, proof= trace the program. The Entscheidungsproblem (descision problem): is there an algorithm that takes as input a statement of a first-order logic and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms. 13

14 A Brief History of Computing Gödel: no! there is no such algorithm Church: better way to prove it with functional programming 14

15 Alan Turing British mathematician cryptanalysis during WWII arguably, father of AI, CS Theory several books, movies Mathematically defined computation 15 Invented Turing Machines at 23 (1936). Turing machines can compute everything that is computable. He proved that The Halting Problem has no general algorithm (it is not possible to decide whether a turing machine will ever halt)

16 Computation Computers were people at that time! The way people do math is write-erase-throw away. Turing proposed to abstract this process. 16

17 Turing Machine

18 Turing Machine Finite alphabet Read Write Move +1 or -1 Halt condition

19 Turing Machine Finite number of internal states Finite alphabet Read Write Was designed as a model of human computation but it models computers as we know them Move +1 or -1 Halt condition Internal state (finite number)

20 Why do we care? I need a model of computation that is simple enough to convince you that a TM can take as input a description of a TM and simulate it. Python interpreter in Python? Harder to explain Python semantics Will be able to define what can we compute?

21 TM for Decision Problems M = (Q, Σ, Γ, B, δ, q start,, q accept, q reject ): Γ is a finite tape alphabet. - B or is the blank symbol (special symbol) - Σ is a finite input alphabet Σ Γ\ B Q is a finite set of states q start Q is the initial state 21 q accept, q reject Q accept/reject states Or maybe run forever

22 TM for Decision Problems M = (Q, Σ, Γ, B, δ, q start,, q accept, q reject ): Γ is a finite tape alphabet. - B or is the blank symbol (special symbol) - Σ is a finite input alphabet Σ Γ\ B Q is a finite set of states q start Q is the initial state q accept, q reject Q accept/reject states 22 Or maybe run forever Transition function: δ : Q Γ (read) Q Γ (write) { L, R }

23 Turing Machine pink tape = string in Γ* followed by infinite stream of we will treat = configuration = state, string (content of tape), and integer (position of tape). (Q,x,i) Q x Γ*x N e.g. (pink, 01011, 5)

24 Configuration = ID (Instantaneous 24 Description) Contains all necessary information to capture the current configuration of the computation state, tape-contents & head-location Easy-to-read notation: (q,xay,i) x Γ* : tape contents left of the head q Q : state y Γ* : tape contents at & right of the head (till last non-blank) Initial ID: (qstart, input,0) a: symbol TM is about to read

25 Relations & * on IDs ID1 ID2 iff ID1 evolves into ID2 in one step. e.g., if δ(q, ai) = (qʹ,b, L), then q,a1 a2 ai-1 ai ai+1 an, i) (qʹ,a1 a2 ai-2ai-1 b ai+1 an,i-1) current ID next ID * i s the reflexive & transitive closure of 25 Thus, ID 1 * ID 2 iff M, when run from ID 1, reaches ID 2 after some finite number (0 or more) of moves

26 Accept/Reject In DFAs it was clear when to accept an input string or reject it. TM can: go back and forth over the input overwrite the input write on the tape way past the end of the input need an explicit state

27 Definition of Acceptance M accepts w iff (qstart,w, 0) * (qaccept,x,i) for some x Γ* Note that M is allowed to accept w without scanning all of w L(M) = {w M accepts w} M does not accept w if starting from the ID qstart w : 1. M halts in qreject, or 2. M crashes (head moves off the tape), or 3. M never stops 27

28 Deciding/Recognizing a Language L(M) = {w M accepts w} is called the language recognized by M M decides L(M) if on input w L, M halts in qreject If a TM decides the language it recognizes, then, on every input, it halts in qaccept or qreject Easy to change crashes to rejects But turns out the we can t avoid infinite executions! (can t tell if it is going to be infinite) 28

29 Deciding/Recognizing a Language L(M) = {w M accepts w} is called the language recognized by M M decides L(M) if on input w L, M halts in qreject Fundamental questions of computability: Which languages are recognizable? Which languages are decidable? Recursively Enumerable Language Recursive Language 29

30 Example Input alphabet : Σ = {0,1} Tape alphabet : Γ = {0,1,0ʹ, 1ʹ, _} What does this TM do? 0 / 0, R 1 / 1, R 0 / 0ʹ, R 1 / 1ʹ, R 0 / 0, L 1 / 1, L _ / 0, L q3 0ʹ / 0, R q1 _ / _, L q2 1ʹ / 1, R 0 / 0, R 1 / 1, R _ / 1, L q4

31 q1 q2 q4 q2 q3 q2 Example ʹ 0 1 0ʹ 0ʹ 1 0ʹ 0ʹ 1ʹ 0ʹ 0ʹ 1ʹ 0ʹ 0ʹ 1 0ʹ 0ʹ 1 1 _ 0ʹ 0ʹ 1 1 _ 0ʹ _ 0ʹ _ 0ʹ _ 0ʹ ʹ ʹ ʹ Input alphabet : Σ = {0,1} Tape alphabet : Γ = {0,1,0ʹ, 1ʹ, _} 0 / 0ʹ, R 1 / 1ʹ, R q1 _ / _, L 0 / 0, L 1 / 1, L q2 _ / 0, L 0ʹ / 0, R 1ʹ / 1, R _ / 1, L 0 / 0, R 1 / 1, R q3 0 / 0, R 1 / 1, R q4

32 q1 q2 q3 q2 Example ʹ _ _ _ _ _ Maps w to ww R Next? Crashes! Head moves out of the tape. Input alphabet : Σ = {0,1} Tape alphabet : Γ = {0,1,0ʹ, 1ʹ, _} 0 / 0ʹ, R 1 / 1ʹ, R q1 _ / _, L 0 / 0, L 1 / 1, L q2 _ / 0, L 0ʹ / 0, R 1ʹ / 1, R _ / 1, L 0 / 0, R 1 / 1, R q3 0 / 0, R 1 / 1, R q4

33 What does this TM do? 33

34 What does this TM do? 0/0,R 0/$,R p0 _/0,L pstart 1/0,R 0/1,R phalt 34 1/$,R p1 _/1,L 1/1,R

35 q1 q2 q3 q2 Example ʹ _ _ _ _ _ Maps w to ww R Next? Crashes! Head moves out of the tape. Input alphabet : Σ = {0,1} Tape alphabet : Γ = {0,1,0ʹ, 1ʹ, _} 0 / 0ʹ, R 1 / 1ʹ, R q1 _ / _, L 0 / 0, L 1 / 1, L q2 _ / 0, L 0ʹ / 0, R 1ʹ / 1, R _ / 1, L 0 / 0, R 1 / 1, R q3 0 / 0, R 1 / 1, R q4

36 Avoiding Crashing Given M (that may crash), an equivalent Mʹ which goes to qreject instead of crashing Idea: Rewrite input w to be $w, place the head on the first symbol of w and run M. If head ever scans $, move to qreject (and move the head right) 0/$,R p0 0/0,R _/0,L 0/0,L 1/1.L $/$,R pstart 1/0,R 0/1,R p2 q0 36 1/$,R p1 _/1,L 1/1,R

37 Shifting by k Positions Can do shift-by-1 k times. But k scans of tape. To shift by k positions to the right in a single scan: Remember last k symbols. Overwrite current cell with symbol from k cells behind _/$,R pʹʹ0 pstart 0/$,R 1/$,R p0 p1 0/$,R 0/0,R 1/1,R 1/$,R 1/$,R p00 p11 1/$,R 1/0,R 1/0,R 0/1,R 0/1,R 0/0,R p01 p10 1/1,R _/0,R _/0,R _/1,R _/1,R pʹ0 pʹ1 _/0,R _/0,R _/1,R _/1,R 0/0,L 1/1.L $/$,R 37 _/$,R pʹʹ1

38 Binary Addition L = { x#y#z x, y, z {0,1}*, x = y = z, x+y=z in binary } Plan: shift check LSB : 0 1 # 0 1 # 1 0 _ : $ 0 1 # 0 1 # 1 0 : $ _ carry c=0 carry c=1 38 check MSB check finished : @ carry c=0

39 Binary Addition L = { x#y#z x, y, z {0,1}*, x = y = z, x+y=z in binary } Shift input w to make it $w. Scan the tape to ensure w matches (0+1)*#(0+1)*#(0+1)* Return head to the left end (right of $) (In finite memory, carry-bit c initialized to 0) Repeat copy the digit to the left of first # into finite state, and overwrite it with # (replace old # If no digit there, accept if carry is 0 & no digits left; else reject. copy the digit to the left of second # into finite state, and move # left (replace old # If no digit there, reject. check if the right most digit is correct. Reject if no digit or if it is not correct; else erase digit and update carry. Move head to the left end (right of $) 39

40 Verify finished & Accept x=1 Shift & Format Check $/$,R 0 / 0, R 1 / 1, R q0 $/$,R #/@, L 1 /#,R x=0 0 R 0 / 0, R 1 / 1, R #/@, L 1 /#,R 0 R 0 / 0, R 1 / 1, R _/_, L y=1 y=0 $/$,R 0 / _, L c=0 # / #, L 0 / 0, L 1 / 1, L 40 c=1

41 What can a TM do? Can shift by any number i, by keeping 2 i states to remember what were in the first i places of the tape. Can do simple arithmetic operations, addition, multiplication etc. 41

42 What can a TM do? L={ 0 i 1 j 0 k i=j=k } is not a CFL! Can be decided by TM (see notes) 42

43 What can a TM do? A TM can do everything that can be done in a standard programming language (and vice versa) 43